Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?
Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?
Can you explain the strategy for winning this game with any target?
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
It starts quite simple but great opportunities for number discoveries and patterns!
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
Can you explain how this card trick works?
The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.
The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?
Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.
We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Can you find sets of sloping lines that enclose a square?
Here are two kinds of spirals for you to explore. What do you notice?
Think of a number, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I. . . .
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
Find out what a "fault-free" rectangle is and try to make some of your own.
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
Delight your friends with this cunning trick! Can you explain how it works?
In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.
Nim-7 game for an adult and child. Who will be the one to take the last counter?
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?
This challenge asks you to imagine a snake coiling on itself.
You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .
This activity involves rounding four-digit numbers to the nearest thousand.
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
Strike it Out game for an adult and child. Can you stop your partner from being able to go?
Are these statements relating to odd and even numbers always true, sometimes true or never true?
Got It game for an adult and child. How can you play so that you know you will always win?
Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.
An investigation that gives you the opportunity to make and justify predictions.
Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?
If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.
Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?
It would be nice to have a strategy for disentangling any tangled ropes...
Can you tangle yourself up and reach any fraction?
A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.
This task follows on from Build it Up and takes the ideas into three dimensions!
Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
It's easy to work out the areas of most squares that we meet, but what if they were tilted?